Permutation matrices and matrix equivalence over a finite field

Abstract

Let F=GF(q) denote the finite field of order q and Fm×n the ring of m×n matrices over F. Let 𝒫n be the set of all permutation matrices of order n over F so that 𝒫n is ismorphic to Sn. If Ω is a subgroup of 𝒫n and A, BϵFm×n then A is equivalent to B relative to Ω if there exists Pϵ𝒫n such that AP=B. In sections 3 and 4, if Ω=𝒫n formulas are given for the number of equivalence classes of a given order and for the total number of classes. In sections 5 and 6 we study two generalizations of the above definition.